Answer
$$\int_{0}^{1}\frac{3x^{2}+1}{x^{3}+x^{2}+x+1}dx=\frac{5}{2}ln2-\frac{\pi}{4}$$
Work Step by Step
$$\int_{0}^{1}\frac{3x^{2}+1}{x^{3}+x^{2}+x+1}dx=\int_{0}^{1}(\frac{2}{x+1}+\frac{x-1}{x^{2}+1})dx$$
$$=\int_{0}^{1}(\frac{2}{x+1}+\frac{x}{x^{2}+1}-\frac{1}{x^{2}+1})dx$$
$$=\int_{0}^{1}(\frac{2}{x+1}-\frac{1}{x^{2}+1})dx+\frac{1}{2}\int_{0}^{1}\frac{1}{x^{2}+1}d(x^{2})$$
$$\left [2ln(x+1)-arctan\,x\right ]_{0}^{1}+\left [\frac{ln(x^{2}+1)}{2}\right ]_{0}^{1}=\frac{5}{2}ln2-\frac{\pi}{4}$$